Maple's ODE and PDE solvers, nowadays powerful tools for finding exact
solutions, were developed as part of a computational environment for
differential equations (DEs). In this talk, an overview of this
computational environment, focusing on its research and educational
tools, is given. Finally, the present status of Maple's ODE and PDE
solvers, as well as their possible evolution as artificial intelligence
systems, are discussed.

The package NODES (for Numerical Ordinary Differential Equations)
developed in Maple by Larry Shampine and myself is quite convenient
for classroom explorations within Maple. This talk will explore
several application modules, ranging from simple introductory examples
through IVP that require custom Maple tools to generate the input for
NODES. Other tools, useful for assessment of the quality of the
computed solution and its relevance to the problem at hand, are also
discussed.

When certain seemingly innocent computations performed with computers
are not merely wrong, but are dramatically wrong, they provide a
powerful reason to be cautious about computations using computers.
Furthermore, should these calculations be such that they can provide a
general insight into how and when computer calculations do go wrong,
beyond just dismissing error as ``noise'', a natural lifelong
appreciation for the subject of numerical analysis can be engendered
and users exposed to the calculations will become masters of their
tools. The presentation can be regarded as a gallery of simple
calculations that can be depended on to produce such insightful
numerical mayhem, even across different machines and across very
different software. While forms of these numerical ``monsters'' are
not unknown, even experienced computer users can find them capable of
surprises in the potent forms presented.

Divided differences of vector functions can be applied to generate
numerical update formulas for ODEs, as proposed by W. Kahan. A
computer algebra system renders the symbolic manipulations practical to
perform. The technique is presented here using the Maple system. Some
examples illustrate the potential of this hybrid symbolic-numeric
approach to solving initial-value problems. We discuss this in an
educational context.